October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits REGULARITY OF PSEUDOGROUP ORBITS PETER J. OLVER ¤ School of Mathematics University of Minnesota Minneapolis, MN 55455, USA E-mail: [email protected] JUHA POHJANPELTO Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA E-mail: [email protected] Let G be a Lie pseudogroup acting on a manifold M. In this paper we show that under a mild regularity condition the orbits of the induced action of G on the bundle J n(M; p) of nth order jets of p-dimensional submanifolds of M are immersed submanifolds of J n(M; p). 1. Introduction Lie pseudogroups, roughly speaking, are in¯nite dimensional counterparts of local Lie groups of transformations. The ¯rst systematic study of pseu- dogroups was carried out at the end of the 19th century by Lie, whose great insight in the subject was to place the additional condition on the local transformations in a pseudogroup that they form the general solution of a system of partial di®erential equations, the determining equations for the pseudogroup. Nowadays these Lie or continuous pseudogroups play an important role in various problems arising in geometry and mathemat- ical physics including symmetries of di®erential equations, gauge theories, Hamiltonian mechanics, symplectic and Poisson geometry, conformal ge- ometry of surfaces, conformal ¯eld theory and the theory of foliations. Since their introduction a considerable e®ort has been spent on develop- ¤Work partially supported by grant DMS 01-03944 of the National Science Foundation. 1 October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits 2 ing a rigorous foundation for the theory of Lie pseudogroups and the invari- ants of their action, and on their classi¯cation problem, see e.g. Refs. [5], [6], [7], [8], [9], [11] and the references therein. More recently, the au- thors of the paper at hand have employed a moving frames construction [3], [4] to establish a concrete theory for Lie pseudogroups amenable to practical computations. As applications, a direct method for uncovering the structure equations for Lie pseudogroups from the determining equa- tions for the in¯nitesimal generators of the pseudogroup action is obtained (see, in particular, the work [1] on the structure equations for the KdV{ and KP{equations) and systematic methods for constructing complete sys- tems of di®erential invariants and invariant forms for pseudogroup actions are developed. Moreover, the new methods immediately yield syzygies and recurrence relations amongst the various invariant quantities which are in- strumental in uncovering their structure, the knowledge of which is piv- otal e.g. in the implementation of Vessiot's method of group splitting for obtaining explicit noninvariant solutions for systems of partial di®erential equations. Let G be a Lie pseudogroup (a precise de¯nition will be given in Sec. 2) acting of a manifold M. The action of G on M naturally induces an action of G on the extended jet bundle J n(M; p) of nth order jets of submanifolds of M by the usual prolongation process. Our goal in this paper is to prove that under a mild regularity condition on the action of the pseudogroup G on M the orbits of G in J n(M; p) are immersed submanifolds for n su±- ciently large. We were originally lead to the problem in connection of the research reported in Ref. [4] and the result is of importance in the the- oretical constructs therein. Interestingly, as we will see, the submanifold property of G orbits in J n(M; p) is closely related to local solvability of the determining equations for the in¯nitesimal generators of the pseudogroup action on M. The proof of our main result relies on classical work [12] on the structure of the orbits of a set of vector ¯elds originally arising in the study of the accessibility question in the context of control theory. In Sec. 2 we cover some background material on Lie pseudogroups and discuss the regularity condition for pseudogroup actions needed in our main result. Sec. 3 is dedicated to the proof of the submanifold property of orbits of the action of G on the extended jet bundles J n(M; p). October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits 3 2. Tameness of Lie Pseudogroups Let D = D(M) denote the pseudogroup of all local di®eomorphisms of a (n) (n) manifold. We write jz ' for the nth order jet of ' 2 D at z and : (n) (n) n D ! M for the associated jet bundle, where (jz ') = z stands for the (n) (n) (n) n source map. We furthermore write ¿ : D ! M, ¿ (jz ') = '(z) = Z for the target map. De¯nition 2.1. A subset G ½ D is called a pseudogroup3 acting on M if (1) the restriction idjO of the identity mapping to any open O ½ M belongs to G; (2) if ', Ã 2 G, then also the composition ' ± Ã 2 G where de¯ned; (3) if ' 2 G, then also the inverse mapping '¡1 2 G. A pseudogroup G is called a Lie pseudogroup if, in addition, there exists N ¸ 1 so that the following conditions are satis¯ed for all n ¸ N: (4) G(n) ½ D(n) is a smooth, embedded subbundle; n+1 (n+1) (n) (5) ¼n : G ! G is a bundle map; (n) (6) a local di®eomorphism ' of M belongs to G if and only if z ! jz ' is a local section of (n) : G(n) ! M; (7) G(n) = prn¡N G(N) is obtained by prolongation. We call the smallest N satisfying the above conditions the order of the pseudogroup, and unless otherwise speci¯ed, we will assume that n ¸ N in what follows. Note that by (1) and (2), the restriction 'jO of a transfor- mation ' 2 G to any open subset O of the domain of ' is again a member of the pseudogroup. Fix local coordinates (z1; : : : ; zm) about some p 2 M, and let (z; Z) = (z1; : : : ; zm; Z1; : : : ; Zm) denote the induced product coordinates about (p; p) 2 D(0) = M £ M. Due to conditions (4) and (6) above, pseudogroup transformations are locally determined by a system (n) F®(z; Z ) = 0; ® = 1; : : : ; k; (1) of partial di®erential equations, the determining equations for G. Here n ¸ N is ¯xed and (z; Z(n)) stands collectively for the coordinates of D(n) induced by (z; Z). By De¯nition 2.1 the above equations are locally solv- (n) (n) able, that is, given a jet go = (zo; Zo ) satisfying (1), then there is a (n) (n) solution ' 2 G of the equations so that jzo ' = go . Let X = X (M) denote the space of locally de¯ned vector ¯elds on M. Thus the domain U(v) ½ M of v 2 X is an open subset of M. The vertical October 11, 2004 14:39 Proceedings Trim Size: 9in x 6in orbits 4 lift V(n) of a vector ¯eld v 2 X to D(n) is the in¯nitesimal generator of the (n) (n) local one-parameter group ©t of transformations acting on D de¯ned (n) (n) (n) by ©t (jz ') = jz (©t ± '), where ©t stands for the ow map of v. Note that the domain of V(n) is ¿ (n)¡1(U(v)). m a m a v v a V V a Pick = Pa=1 (z)@z 2 X and write = Pa=1 (Z)@Z for the vertical counterpart of v. Then V(n) is simply given by the usual prolongation formula2, m (n) (n) a (n) V z; Z D V z; Z @ a ; ( ) = X X ( J )( ) ZJ (2) a=1 jJj·n where J = (j1; : : : ; jp) is stands for a multi-index of integers, DJ = j Dj1 ¢ ¢ ¢ Djp for the product of the total derivative operators Dj = @z + m a a Z @ a , and where the Z denote the components of the ¯ber Pa=1 PjJj¸0 Jj ZJ J (n) (n) coordinates on D induced by (z; Z). In particular, at the nth jet Iz of the identity mapping, Eq. (2) becomes m (n) a V @ J v z @ a ; (n) = X X z ( ) ZJ (3) Iz a=1 jJj·n where we have again used the obvious multi-index notation. We denote the space of n jets of local di®eomorphisms with source at a (n) (n) ¯xed z 2 M by D jz. It is easy to see that D jz is a regular submanifold (n) (n) (n) of D . Write Rh(n) for the right action of a jet h 2 D on the source (n) (n)¡1 (n) (n) (n) n ¯ber D j¿ (n)(h(n)) = (¿ (h )) by Rh(n) g = j(n)(h(n))(' ± Ã), (n) n (n) n where h = j(n)(h(n))Ã, g = j¿ (n)(h(n))'. Then, by di®erentiating the identity (n) (n) (n) (n) Rh(n) ©t (g ) = ©t (Rh(n) g ); (n) it is easy to see that V is Rh(n) -invariant, that is, (n) (n) (n) (n) Rh(n) ¤(V (g )) = V (Rh(n) g ); (4) (n) (n) (n) (n) whenever (g ) = ¿ (h ). Note that the action of Rh(n) on V(n)(g(n)) is well de¯ned since V(n) is a vertical vector ¯eld. Next let G be a Lie pseudogroup. A local vector ¯eld v 2 X on M is a G vector ¯eld if its ow map ©t is a member of G for all ¯xed t on some interval about 0.